Step by step
The question
Question: find the area A of the sector of a circle with radius 6 and angle π/4.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 6.
And the angle: θ = π/4.
1 Write the rule
The rule for the area of a sector: A = ½r²θ.
θ must be in radians.
Why?
A full turn is 2π.
Then A is a half times r squared times 2π: π r squared, the whole circle.
A smaller angle gives a part of it.
Careful!
r is squared: 6² = 36, not 6.
Area always has a square in it.
2 Put the numbers in
Now put the numbers in: r = 6 and θ = π/4.
6 squared is 36.
3 Work it out
Tops times tops: 36 × π is 36π.
Bottoms: 2 × 4 is 8.
36 and 8 can both be divided by 4.
36 ÷ 4 is 9.
8 ÷ 4 is 2.
So A = 9π/2 square units.
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 14.14 square units, to the nearest hundredth.
4 Check
Let's check another way.
π/4 is a part of the full turn, 2π: 1/8 of it.
The whole circle is π r squared: 36π.
1/8 of 36π is 9π/2.
The same answer!
So it is right.
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.